Fix $T > 0$ and let $t \in (0,T)$ let $c > 1$ be a constant (which may be bigger than $T$).
Consider the function $$f(c,t,T) = \frac{\sinh ((T-t)c)}{\sinh (Tc)}.$$ I am looking for a bound of the form $$f(c,t,T) \leq {K(t,T)}{c^{-\frac 12}} \qquad \text{for $t \in (0,T)$}$$ where $K(t,T)$ is a constant that depends on $t$ and $T$ (preferably in a continuous or a nice way). Can such a bound be possible?
By using half-angle formula, we can write $$f(c,t,T)=\cosh(tc)-\coth(Tc)\sinh(tc)$$ but I just don't see how to estimate it.